## The difference between selection coefficient and relative growth rate

If you do experiments on microorganisms, you are probably familiar with fitness assays, where you study a mutant strain by comparing its growth rate to that of the wildtype in various environments. If, like me, you have learned the method by reading papers, you may have missed an important fact: there are two different ways of presenting fitness data. Microbial fitness can be reported as either a selection coefficient or a relative growth rate difference, and although these are mathematically related, they are not equivalent. Moreover, they have different conceptual meanings. This point may be obvious to some, but is subtle enough that I thought the two quantities were equal until I had read (and failed to understand) multiple explanations to the contrary.

Here I will try to help other confused experimentalists by explaining this difference in practical terms, as it arises in the analysis of a competitive growth assay. None of this material is original, but I hope my presentation clarifies ideas that aren’t as transparent elsewhere. I borrow especially heavily from Greg Lang’s competition assay protocol, the Crow and Kimura population genetics textbook, and this article by Luis-Miguel Chevin that makes the same point I make here, but at a more conceptual level.

If you’re the impatient type, everything below can be summarized in 3 points:

• A 1% difference between two strains in the number of offspring per generation (selection coefficient) is not the same as a 1% difference in exponential growth rate (relative growth rate difference).
• A 1% difference in number of offspring per generation is approximately equal to a 0.69% difference in exponential growth rate.
• You should report fitness as a selection coefficient if you can. If you must use a different metric, state this prominently so people like me don’t get confused.

### The experimental design

In a competitive growth assay, a “query” strain is cultured in the same tube as a “reference” strain and the relative abundance of the two types of cells is monitored over time. One or both of the strains is marked by a fluorescent protein (or traditionally, a drug/nutritional marker) to allow the co-cultured cells to be distinguished and counted. The goal is to compute some metric that represents the difference in fitness of the query strain with respect to the reference strain. When multiple query strains are co-cultured with a common query strain, the fitnesses of the query strains can be compared to each other.

### Relative fitness as selection coefficient

The standard metric computed from competition assays is the selection coefficient $s$, which represents the relative difference in number of offspring per generation between a mutant and wildtype organism. For example, if every wildtype individual becomes 3 individuals in the next generation (e.g. by giving birth to 2 additional individuals, or giving birth to 3 individuals and then dying), and each mutant individual becomes 2 individuals in the next generation, then the selection coefficient of the mutant with respect to the wildtype is $s=2/3-1=-0.333$. This is often reported as a “33% per generation” fitness disadvantage. The selection coefficient is a fundamental concept in population genetics because it provides a way to quantify the evolutionary dynamics of mutations while abstracting away the underlying details of birth rate, death rate, generation time, etc.

To write $s$ in terms of quantities measured in a competitive growth experiment, denote the ratio of the number of query strain cells to the number of reference strain cells as $R=N_q/N_r$. At the beginning of the experiment, $R=R(0)$. After 1 generation of growth, $R(1)=R(0)(1+s)$. After 2 generations of growth, $R(1)=R(0) (1+s)^2$. After $\tau$ generations of growth,

$\displaystyle R(\tau)=R(0) (1+s)^\tau.\ \ \ \ (1)$

Taking the natural logarithm of this, we get

$\displaystyle \log_e R(\tau)-\log_e R(0)=\tau \log_e (1+s)$

If the fitness difference is small (i.e. $s<0.05$), which is usually true if we’re going to the trouble of doing a competition assay, we can use the approximation $\log_e (1+s) \approx s$. This allows us to rewrite the above to get

$\displaystyle s=\frac{\Delta \log_e R}{\Delta\tau}$

Here, $\Delta \log_e R = \log_e R(\tau)-\log_e R(0)$ is the change in log-ratio of query-to-reference cell number and $\Delta\tau$ is the number of generations, or “doublings” of the microbial culture, during some time interval. Since the choice of when $\tau=0$ is arbitrary (i.e. we could just as well use $\tau_1$ and $\tau_2$ as our start and end times), I will use the $\Delta$ notation from now on. Since the query and reference strains generally have different doubling times, we’ll define the generation time, by convention, as the doubling time of the reference strain. Therefore, the number of generations is just the log-base-2 of the fold-change in reference-strain cell count, or $\Delta\tau = \log_2 \left ( N_r (\tau)/N_r (0) \right) = \Delta \log_2 N_r$, and

$\displaystyle s=\frac{\Delta \log_e R}{\Delta \log_2 N_r}. \ \ \ \ (2)$

How does this help us determine $s$ from data? $R$ can be measured by colony counting or flow cytometry. To determine the denominator, we can measure the optical density $OD$ of the culture, which is proportional to the total cell count: $OD \propto N_q + N_r$. This means that $N_r \propto \frac{1}{R+1} OD$ (i.e. the number of reference strain cells is proportional to the fraction of the optical density contributed by the reference strain). Therefore,

$\displaystyle \Delta \tau = \Delta \log_2 \left ( \frac{1}{R+1} OD \right )$.

These formulas allow estimation of $s$ using measurements of the strain ratio and optical density at just 2 timepoints. Alternatively, optical density doesn’t even need to be measured if the culture starts at a known dilution from saturation and ends at saturation. Then, the number of doublings of the culture is the $\log2$ of the inoculating dilution factor. Some studies studies take multiple time points and calculate $s$ by fitting a line to a plot of $\log R$ versus $\tau$. If $s$ is constant throughout the experiment, these data should fit tightly to a straight line. If the line fit is poor, it may suggest strain differences in lag time or other transient effects, and motivate some data filtering or protocol optimization.

#### Relative fitness as relative growth-rate difference

In defining the selection coefficient above, we thought about population growth over discrete generations. This is a natural model for organisms that have synchronized reproductive cycles and can differ in the number of offspring per generation. However, microbes reproduce by cell division, which always converts 1 cell into 2, and differences in fitness are usually due to different generation times. Therefore, it can be more intuitive to think about microbial fitness in terms of continuous growth. Imagine that the query and reference strains grow (i.e. $N_q$ and $N_r$ increase) exponentially with growth rates (or “Malthusian parameters”) $m_q$ and $m_r$. This means that

$\displaystyle \begin{array} {ccc} N_q(t) & = & N_q(0)\cdot e^{m_q t} \\ N_r(t) & = & N_r(0)\cdot e^{m_r t}. \\ \end{array} \ \ \ \ (3)$

Here, $t$ represents “clock” time, let’s say hours. Then, the growth rates are in units of 1/hour (or “$e$-fold increases per hour”). The strain ratio $R$ is the ratio of the above expressions, or $R(t) = R(0)\cdot e^{(m_q-m_r )t}$. Taking the natural $\log$ again, we get

$\displaystyle \log_e R(t) - \log_e R(0)=(m_q-m_r )t$.

In the notation from above, this rearranges to

$\displaystyle m_q-m_r=\frac{\Delta \log_e R}{\Delta t}. \ \ \ \ (4)$

This equation shows that on a plot of $\log_e R$ versus $t$ (in hours), the slope represents the growth rate difference (sometimes called “selection rate” and denoted by $r$) between the strains. The similarity of equation (4) to equation (2) suggests an analogy: whereas the selection coefficient $s$ represents the fitness difference over “generation time” units, we can think of the growth rate difference $m_q-m_r$ as representing the fitness difference in “clock time” units. In this sense, though, it seems that $s$ has an appealing property that $m_q-m_r$ doesn’t have, which is invariance to the absolute growth rates of strains. However, we can get this feature if we normalize the growth rate difference by the reference strain growth rate. Let’s call this result the relative growth rate difference $s_{GR}$:

$\displaystyle s_{GR}=\frac{m_q-m_r}{m_r}. \ \ \ \ (5)$

This quantity seems truly analogous to $s$. Whereas $s=0.01$ can be thought of as the query strain producing 1% more offspring per generation than the reference strain, $s_{GR}=0.01$ represents the query strain having a 1% faster growth rate than the reference strain.

### Selection coefficient is not equal to relative growth rate difference

At this point, you might be tempted to conclude on intuitive grounds (as I did) that once normalized relative to a reference strain, the difference in offspring per generation and growth rate actually amount to the same thing, i.e. $s=s_{GR}$. This is not true. To see this, we can write $s_{GR}$ in a similar form to equation (2). First, use the exponential growth equations (3) to get $m_r = \Delta\log_e N_r /\Delta t$, where $N_r$ can be computed from optical density as explained above. Combining this with definition (4) and equation (5), we get

$\displaystyle s_{GR}=\frac{\Delta \log_e R}{\Delta \log_e N_r}$.

Changing the base of the logarithm in the denominator, we get

$\displaystyle s_{GR} = \log_e 2 \cdot \frac{\Delta \log_e R}{\Delta \log_2 N_r}$.

The last fraction on the right-hand side is equal to $s$ as defined by (1), so the above simplifies to

$\displaystyle s_{GR} = \log_e 2 \cdot s$.

This means that the relative growth rate difference is not equal to the selection coefficient, but rather proportional to it by a factor of $\log_e 2 \approx 0.69$. Therefore, the example of $s=0.01$, where a query strain produces 1% more offspring per generation than the reference strain, is equivalent to $s_{GR}=0.0069$, or the query strain having a 0.69% faster growth rate than the reference strain.

This distinction may be obvious at this point. It boils down to algebraic properties of the exponential function—the number of offspring appears in the base of an exponential (equation (1)) whereas growth rate is in the exponent itself (equation (3)), so relative differences in these quantities should not be mathematically (or conceptually) equivalent. It isn’t even true in general that $s$ is proportional to $s_{GR}$—this just happens to hold when $s$ is small and the approximation $\log_e (1+s)=s$ can be used.

### Which metric to use?

Based on the literature I have read, most papers report fitness measurements using the selection coefficient $s$. When authors have explicitly discussed the two approaches, they advocate using $s$ because it is easily defined for any organism (“growth rate” isn’t an intuitive concept for, say, zebras) and draws a direct parallel to population genetics theory. On the other hand, “number of offspring” seems an awkward metaphor for differences in doubling times of microbes. Growth rate is arguably a better description of actual microbial physiology. Presumably this is why a minority of papers, mostly in microbial evolution, report fitness using (relative) growth rates instead of selection coefficients. However, I would hew to Chevin’s suggestion to report selection coefficients if at all possible, since it is by far the standard practice, and the slight conceptual awkwardness doesn’t prevent $s$ from being easily computed.

Sometimes, there is a methodological reason to avoid the selection coefficient. For example, if fitness is measured by growth curves (culture density over time), then growth rate (i.e. $m = \frac{d}{dt} \log OD$) is the only reasonable metric to compute. To compare strains in this context, the natural thing to do is to normalize growth rates by defining $s_{GR} = m_q/m_r -1$ or simply $m_q/m_r$ , rather than introducing another factor of $\log_e 2$, which may confuse readers. There are also unusual situations where normalizing by a reference strain is impossible or confusing–for example, if strains are not growing but dying and thus $m_r$ is negative, or if relative growth rates change dynamically throughout the experiment. In these cases, the growth rate difference $m_q-m_r$ can be a more convenient metric.

My suggestion, if you do not use the selection coefficient, is to include a note explicitly describing whether and how your results can be converted to selection coefficients, as well as any supplemental data necessary to do this (i.e. the reference strain growth rates). And in all cases, clearly define the metric you use and avoid “overwriting” existing nomenclature with non-standard definitions.

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